So for the definition of Dehn surgery (also called rational/integer surgery), what is correct:

Definition
Let $K$ be a knot in an oriented $3$-manifold with a regular neighbourhood $N(K)\simeq S^1\times D^2$. Dehn surgery is the operation of removing $\operatorname{int} N(K)$ and gluing in $S^1\times D^2$ by an arbitrary / orientation reversing (?) diffeomorphism mapping from the boundary torus $S^1\times \partial D^2$ to the boundary of $N(K)$.